What is Vector: Definition and 1000 Discussions

The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...

For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds: $$ df = \sum^n \frac{\partial f}{\partial x_i} dx_i $$ If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...

What is the notation to show element-wise square root of a vector or matrix?

I calculated force vector by differentiating momentum vector.Since acceleration and velocity vectors are at45°,therefore force and momentum vector are at 45°.But i am not able to find the time at which it will take place.I tried F vector.P vector=FPcos45° but i am not getting from it.I also used...

If within a volume v ,there exists 10 velocity fields at different points then can anyone please suggest how to compute ##\int_v(\nabla•v)## within the volume?? using matlab For exm if the velocity vector field be ##v=x\hat x+y\hat y+z\hat z## and for x=1 to 10,y=1 to 10 and z= 1 to 10 the 10...

To be honest i don't know from where to start. I know how i can test the stokes theorem if i have a cylindrical shape and a cylindrical vector or spherical vector and a spherical shape but here I am out of ideals. The first thing i tried was to compute the left part of the stokes theorem but i...

Hi PF! Given a list of numbers, how do I select the element that has the smallest real part? I don't just want the real part though, I want the entire component. I googled this and tried a few things but nothing works.

https://www.physicsforums.com/attachments/9050 ok I think I got (a) and (b) on just observation but (c) doesn't look like x,y,z will be intergers so ?

Determine if the set of vectors $\begin{bmatrix} x\\y\\5 \end{bmatrix}\in \Bbb{R}^3$ form a vector space ok if I follow the book example I think this is what is done $\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5...

I am trying to update a vector. I have tried this but it's not working. Any ideas? \[Lambda] = {1, 2} For[\[CapitalLambda] = 0.1, \[CapitalLambda] <= 0.2, \[CapitalLambda] += 0.1, \[Lambda] = Catenate[{\[Lambda], {\[CapitalLambda]}}] // Print]

Hi i am e-pie's brother and he let me use his account.Since $$\vec{\omega}$$ is always perpendicular to the plane.Shouldn't this be $$0^o$$?

The magnitude of each force is shown below: F1 = 10 N F2 = 20 N F3 = 40 N R = \sqrt {Rx^2 + Ry^2} R = \sqrt {-10^2 -15^2} = 18N θ = tan^{-1} \frac{Ry}{Rx} θ = tan^{-1} \frac{Ry}{Rx} = 56 To express the direction of R, we need to calculate the direction angle (i.e. the counterclockwise angle...

Summary: I've posted this in a few forums but still confused on this problem. If the plane is moving at 200km/h and the wind pushes the plane with a velocity of 85km/h, then the resultant velocity would be 217km/h, and using sine inverse, 217.sin(theta=85, I got 23°, which is B, but the answer...

Dear everyone. I'm doing an assignment on vectorfields and for most of the assignment I have to deal with tensors and tensornotation. The first assignment asks me to express the following vector and matrixproducts in tensornotation. $$\overline c = \overline a + \overline b \\ d=(\overline a +...

We have the retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'## And its curl ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...

We know that the sum of the two vectors is represented by the major diagonal. Now, if I draw the vector diagram, it turns out to be a diagonal of an entirely different parallelogram. So, my first answer was (c) none. But, I noticed that when parallelly shifted, it turns out to be the minor...

In dealing with rotating objects, I have found the need to be able to transform a vector field from cylindrical coordinate systems with one set of coordinate axes to another set. For eg i'd like to transform a vector field from being measured in a set of cylindrical coordinates with origin at...

The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction. The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current...

Hi, I want to plot the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k## The code I have tried: # The components of the vector field F_x = y*e**x F_y = x**2 + e**x F_z = z**2*e**z# The grid xf = np.linspace(-0.15, 2.25, 8) yf = np.linspace(-0.15, 2.25, 8) zf =...

I am starting to learn classical physics for my own. One exercise was, to calculate the vector r (see picture: 1.47 b). The vector r is r=z*z+p*p. I don’t understand this solution. My problem is: in a vector space with n dimensions there are n basis vectors. In the case of cylindrical...

Is a constant vector field like F = kj conservative? Since the work of F for any closed path is null it seems that F is conservative but for a force to be conservative two conditions must be satisfied: a) The force must be a function of the position. b) The circulation of force is zero. My...

If the curl of a vector is 0 e,g ##\vec \nabla×\vec A=0## the vector A is said to be irrotational,can anyone please tell how rotation is involved with ##curl## of a vector??

I am writing about the nature of force in classical mechanics and what does really imply, in terms of change in motion. I am using as an example a circuit, on which we exert a force. I am trying to justify the following scheme (concretely, ##f_{mag}##): The thing is that I am wondering how...

I have a question about a rotating vector field: if there is a vector ##A^i(t_0)## at the origin in coordinate space ##IR^3## , when ##t≥t_0##, the vector rotates with a changing angular-velocity ##ω^i(t)##. then we can get a rotating vector field ##A^i(t)##. then how to describe ##A^i(t)##...

So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following \frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}...

Why are the red circled Del operators not combining to become 'Del-squared' to cancel out the second term to give a net result of 0?

While I was going through "Introduction to Electrodynamics" by David J. Griffith I see the line "Current is a vector quantity". But we know it doesn't obey the vector algebra (addition ). Then how it can be a vector?... Please help me

Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...

Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...

We commonly have E and B defined as: But how can I work in electric field E, instead of vector potential A?

Hello everybody. The Lagrangian for a massive vector field is: $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^2}{2}A_\mu A^\mu$$ The equation of motion is ##\partial_\mu F^{\mu\nu}+m^2A^\nu = 0## Expanding the EOM with the definition of ##F^{\mu\nu}## the Klein-Gordon equation for...

Hi, I'm trying to find all the valid surfaces that go through a vector field so that the normal of the surface at any point is equal with the vector from the vector field at the same point. The vector field is defined by the function: $$ \hat N(p) = \hat L(p) \cos \theta + \hat R(p)...

The difference between light and very heavy atoms reflects itself in these two schemes. My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ? In other words, why & how the relative magnitude of the...

Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...

I don't understand, please ckeck Let V=\Bbb{R}^2 and {u=(u_1,u_2), v=(v_1.v_2)}\in\Bbb{R}^2 , {k}\in \Bbb{R} define of operation u\oplus v = (u_1+v_1,u_2+v_2) and k \odot u =(2ku_1,2ku_2) check V is vector over field \Bbb{R} ? ________________________________________________________________ I...

Homework Statement The diagram shows parallelogram ABCD. (you don't really need the diagram) vector AB= (2 above, 7 below) and vector AC= (10 above, 11 below) The point B has coordinates (5, 8) (a) Work out the coordinates of the point C. The point E has coordinates (63, 211) (b) Use a...

I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus). If you also have online recommendations please feel free to write them.

I'm trying to figure out this volume integral, a triple integral, of a 9-variable function. 3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates. After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...

I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...

In a recent thread, the following was posted regarding the "no hair" theorem for black holes: In the arxiv paper linked to, it says the following (p. 2, after Theorem 1.1): "Hawking has shown that in addition to the original, stationary, Killing field, which has to be tangent to the event...

Given the following vectors: how can i determine that Θ = Δp/p ? I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...

In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$. But from the transformation of contravariant vector...

In my pre-calculus textbook, the problem states: A 200-pound cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp? The gravitational force can be represented by the vector F=0i-200j In order to find the force we need to project vector...

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...

Let us have some localized density of sources, S, in a plane, each of which produces a localized circular vector field. Let us work in polar coordinates. Let the density of sources, S = Aexp(-r^2/a^2) and let each source have circular vector field whose strength is given by exp(-(r-r_i)^2/b^2)...

Homework Statement Homework Equations v = I + j + k v = d/t The Attempt at a Solution I thought the answer was as simple as: v = 63i + 0j + 0k, since the car only has motion in one direction... ...but I got it wrong, so clearly I'm missing something here.

I use the ##(-,+,+,+)## signature. In the Schwarzschild solution $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$ with coordinates $$(t,r,\theta,\phi)$$ the timelike Killing vector $$K^a=\delta^a_0=\partial_0=(1,0,0,0)$$ has a norm squared of...